Numerical Study of Ion-Slip and Hall Effect on Couette Flow of Two Immiscible Micropolar and Micropolar Dusty Fluid (Fluid-Particle Suspension) with Heat Transfer

Hall current with ion slip has significant roles in various scientific processes including magnetohydrodynamic reactors, refrigeration coils, hall accelerators, flowmeters, and also in-flight magneto-aerodynamics. In the presence of a strong magnetic field, the effect of the Hall current and ion slip is important. As a result, it is essential to include the effect of the magnetohydrodynamics formulation, Hall current, and ion slip terms in the spectrum of functionalities such as a solid-state fuel-powered MHD engine. Inferred from these facts, many studies into MHD flows in the presence of Hall and ion slip currents have been conducted. The effect of ion slip on MHD energy monitoring of plasma flow around a blunt body is investigated by Yoshino et al. [1]. Elahi et al. [2] investigated the peristaltic flow of Jeffrey fluid in a non-uniform rectangular channel with Hall and ion slip effects and discussed the trapping phenomenon of MHD flow. Krishna and Chamkha [3] explored the Hall and Ion-slip implications for the MHD convective flow of viscous liquid through the permeable medium with applied periodic pressure gradient. Goud [4] the effect of heat generation and diffusion on a stretched quasi vertical plate flowing through a porous medium in a steady MHD flow. On the MHD free convective flow of the Casson fluid model, Rajkumar et al. [5, 6] found the effects of radiation absorption and viscous dissipation. Fendoğlu et al. [7] explored the magnetohydrodynamic (MHD) flow of viscous fluid in a rectangular channel with a perplexed boundary.

The advancement of micropolar fluids has arisen as an important area of research for numerous non-Newtonian fluid flow investigations such as blood transfer into vessels, turbulent shear flows, liquid crystals, and lubrication processes. The micropolar liquids are non-Newtonian liquids comprising of polymeric and rotating microstructure or colloidal liquid components like enormous free weight particles. The properties of such non-Newtonian fluids are even not sufficiently described by Navier-Stokes’s equations. Eringen developed the fundamental modeling of micropolar fluid flow [8, 9]. The study of micropolar fluid with temperature distribution has been done by Fakour et al. [10]. The interaction between solid balls rotating through a micropolar fluid has been done by Sherief et al. [11]. The hydrodynamic velocity for quasi micropolar fluid was calculated by Saad et al. [12]. Normal convection was studied by Mehryan et al. [13] within a porous structured platform equipped with micropolar nanofluid. Under the MHD effect, heat transfer analysis through variable viscosity inside a duct comprising micropolar fluid has been investigated by Javed and Siddiqui [14]. Tetbirt et al. [15] Numerical investigation of the magnetic impact on the velocity propagation of a micropolar and viscous fluid in a longitudinal path.

The suspension of tiny (dust particles) in a mixture of nanofluid is normally used to improve the conductivity of the base liquid, called dusty liquid. It can be used for oxidation, petroleum products aeration, and ionic accumulation, among other things. For micropolar dusty free convection flow, Reddy and Ferdows [16] observed energy and momentum transportations. Ghadikolaei et al. [17-19] investigated the immersion of composite nanoparticles in micropolar liquid. Eid and Mabood [20] investigated the deposition of nanotubes in MHD micropolar dusty liquid. Makinde et al. [21] carried out a detailed simulation solution of MHD melting micropolar solution and discovered the heat transfer generation. Attia et al. discovered the effect of different fluid parameters on the velocity and temperature profiles of dusty fluid flow [22].

Numerous problems in hydrology, chemical engineering, oil filtration and recuperation, and other domains may be viewed as processes containing two immiscible fluids. Immiscible liquid streams are used in a variety of applications such as hemodynamics that refers to demonstrate the blood movement through semi-blocked veins. Devakar et al. [23, 24] found the effect of various fluid parameter which are present only in one of the immiscible fluids but also affects the velocity, temperatures of other. Srinivas and Murthy [25-27] reviewed the two immiscible fluid flow in various schemes of different shapes. Umavathi et al. [28-30] have been done a further study on the flow of immiscible fluids between parallel sheets for Newtonian and non-Newtonian fluids.

There are different mathematical techniques for addressing the solution of the partial differential equation. The differential quadrature technique (DQM) is the most noticeable for managing unsteady and nonlinear terms in the coupled equation. Arora and Joshi [31] settled the nonlinear Burgers condition for two spatial dimensions. Ramesh and Joshi [32] simulated unsteady Jeffery liquid flow between two permeable plates utilizing the geometrical B-spline differential quadrature technique. Two immiscible micropolar and Newtonian liquids flow were simulated by the cubic B-Spline differential quadrature technique [33].

Using various shapes and numerical simulations, researchers have conducted several experiments on the steady and unsteady flow of single and immiscible combinations of micropolar, Newtonian fluids, resulting in many results. One of the two-phase flow streams that can be investigated using magnetohydrodynamic and thermal effects is two immiscible micropolar fluids.

In the present work, the unsteady Couette flow of two immiscible micropolar and micropolar dusty (fluid-particle suspension) fluids are considered in the horizontal channel.

The Couette flow of these non-Newtonian, viscous, and incompressible fluids is studied with heat transfer and externally applied magnetic field.

In light of this, we intend to investigate Ion slip, Viscous dissipation, Joule heating, and Hall current effect on flow velocity, microrotation, and temperature profiles. The modeling of the coupled partial differential equation has been done and numerical results are obtained for velocity, microrotation, and temperature profiles under the effect of various important fluid parameters.

To analyze the fluid velocity and temperature distribution for schemes I and II, the domain [-1, 1] is split into [-1, 0] for pure fluid (region –I) and [0, 1] for dusty fluid (region-II). Next both domains are likewise discretized with step length h in the y-(transverse) direction and k’ in the time scales. Similarly, the domain [-1, 1] is uniformly discretized for flow analysis for scheme III. The nodes are presumed to disperse uniformly.

$a=y_{1}<y_{2}<\cdots<y<y_{n}=b$, such that $y_{i+1}-y_{i}=h$ on the real axis.                       (25)

Following this, let the $R_{i_{y}}\left(y_{i}, t\right)$ is the first and $R_{i_{y y}}\left(y_{i}, t\right)$ are the second-order derivatives of $U_{1}(y, t), U_{2}(y, t), U_{P}(y, t), N_{1}(y, t), N_{2}(y, t) T_{1}(y, t), T_{2}(y, t)$ and $T_{P}(y, t)$ are obtained at any time on the nodes y_i.

For $i=1,2,3, \ldots, n . U_{1_{ y}}\left(y_{i}, t\right)=\sum_{j=1}^{N} X_{i j}^{*} U_{1}\left(y_{j}, t\right)$                    (26)

$U_{1_{ y y}}\left(y_{i}, t\right)=\sum_{i=1}^{N} Y^{*} U_{1}\left(y_{j}, t\right)$                      (27)

For $i=1,2,3, \ldots, n . U_{2_{ y}}\left(y_{i}, t\right)=\sum_{j=1}^{N} X_{i j}^{*} U_{2}\left(y_{j}, t\right)$                       (28)

$U_{2_{ y y}}\left(y_{i}, t\right)=\sum_{j=1}^{N} Y_{i j}^{*} U_{2}\left(y_{j}, t\right)$                     (29)

For $i=1,2,3, \ldots, n \cdot U_{p_{y}}\left(y_{i}, t\right)=\sum_{j=1}^{N} X_{i j}^{*} U_{p}\left(y_{j}, t\right)$               (30)

$U_{p_{y y}}\left(y_{i}, t\right)=\sum_{j=1}^{N} Y^{*}{ }_{i j} U_{p}\left(y_{j}, t\right)$                   (31)

For $i=1,2,3, \ldots, n . N_{1_{ y}}\left(y_{i}, t\right)=\sum_{j=1}^{N} X^{*}{ }_{i j} N_{1}\left(y_{j}, t\right)$                 (32)

$N_{1_{ y y}}\left(y_{i}, t\right)=\sum_{j=1}^{N} Y_{i j}^{*} N_{1}\left(y_{j}, t\right)$                   (33)

For $i=1,2,3, \ldots, n . N_{2 _{y}}\left(y_{i}, t\right)=\sum_{j=1}^{N} X_{i j}^{*} N_{2}\left(y_{j}, t\right)$                    (34)

$N_{2_{ y y}}\left(y_{i}, t\right)=\sum_{i=1}^{N} Y_{i j}^{*} N_{2}\left(y_{j}, t\right)$                  (35)

For $i=1,2,3, \ldots, n . T_{1_{y}}\left(y_{i}, t\right)=\sum_{j=1}^{N} X_{i j}^{*} T_{1}\left(y_{j}, t\right)$                    (36)

$T_{1_{ y y}}\left(y_{i}, t\right)=\sum_{j=1}^{N} Y_{i j}^{*} T_{1}\left(y_{j}, t\right)$                     (37)

For $i=1,2,3, \ldots, n . T_{2_{ y}}\left(y_{i}, t\right)=\sum_{j=1}^{N} X_{i j}^{*} T_{2}\left(y_{j}, t\right)$                      (38)

$T_{2_{ y y}}\left(y_{i}, t\right)=\sum_{i=1}^{N} Y^{*}{ }_{i j} T_{2}\left(y_{j}, t\right)$                       (39)

For $i=1,2,3, \ldots, n \cdot T_{p_{y}}\left(y_{i}, t\right)=\sum_{j=1}^{N} X_{i j}^{*} T_{p}\left(y_{j}, t\right)$                   (40)

$T_{p_{y y}}\left(y_{i}, t\right)=\sum_{j=1}^{N} Y^{*}{ }_{i j} T_{p}\left(y_{j}, t\right)$                     (41)

Here $X^{*}{ }_{i j} Y^{*}{ }_{i j}$ are the respective weighting coefficients of 1^st and 2^nd order derivative coefficients concerning y-coordinate, measured using modified cubic B-spline functions. The cubic B-spline functions of the knots are as specified below.

$\theta_{j}=\frac{1}{h^{3}}\left\{\begin{array}{cc}\left(y-y_{j-2}\right)^{3}, & y \epsilon\left[y_{j-2}, y_{j-1}\right) \\ \left(y-y_{j-2}\right)^{3}-4\left(y-y_{j-1}\right)^{3}, &y \epsilon\left[y_{j-1}, y_{j}\right) \\ \left(y_{j+2}-y\right)^{3}-4\left(y_{j+1}-y\right)^{3}, &y \epsilon\left[y_{j}, y_{j+1}\right) \\ \left(y_{j+2}-y\right)^{3}, & y \in\left[y_{j+1}, y_{j+2}\right) \\ 0, & \text { otherwise } .\end{array}\right.$                          (42)

$\theta_{j}$ with $j=0,1,2 \ldots, N+1$ forms a basis over the region [a, b].

The updated cubic B-spline functions are described in the nodes as follows.

$F_{1}(y)=\theta_{1}(y)+2 \theta_{0}(y)$                        (43)

$F_{2}(y)=\theta_{2}(y)-\theta_{0}(y)$                      (44)

$F_{j}(y)=\theta_{j}$, for $j=3, \ldots, N-2$                 (45)

$F_{N-1}(y)=\theta_{N-1}(y)-\theta_{N+1}(y)$                     (46)

$F_{N}(y)=\theta_{N}(y)+2 \theta_{N+1}(y)$                     (47)

Here $\left\{F_{1} F_{2}, \ldots, F_{N}\right\}$ forms a basis over the region [a,b]. The weighting coefficients are derived using the modified cubic B-spline function. The approximation for the 1^st order derivative is:

$F_{k}^{\prime}\left(y_{i}\right)=\sum_{i=1}^{N} X^{*}{ }_{i j} \theta_{k}\left(y_{j}\right) \begin{gathered}\text { for } i=1,2, \ldots, N \\ k=1,2, \ldots, N\end{gathered}$                    (48)

The estimate can be provided for the first-knot point y₁.

$F_{k}^{\prime}\left(y_{1}\right)=\sum_{i=1}^{N} X^{*}{ }_{1 j} \theta_{k}\left(y_{j}\right) \quad \begin{gathered}\text { for } i=1,2, \ldots, N \\ k=1,2, \ldots, N\end{gathered}$               (49)

Then the tri-diagonal system of equations is formed as

$\left[\begin{array}{ccccccccc}6 & 1 & 0 & 0 & & & & & \\ 0 & 4 & 1 & 0 & \cdots & 0 & & 0 & \\ 0 & 1 & 4 & 1 & & & & & \\ & \vdots & & 0 & \ddots & 0 & & \vdots & \\ & & & & & 1 & 4 & 1 & 0 \\ & 0 & & 0 & \cdots & 0 & 1 & 4 & 0 \\ & & & & & 0 & 0 & 1 & 6\end{array}\right]\left[\begin{array}{c}X^{*}{ }_{11} \\ X^{*}{ }_{12} \\ X^{*}{ }_{13} \\ \cdot \\ \cdot \\ \cdot \\ X^{*}{ }_{N-3} \\ X^{*}{ }_{N-2} \\ X^{*}{ }_{N-1} \\ X^{*}_{N} \end{array}\right]=\left[\begin{array}{c}-6 / h \\ 6 / h \\ 0 \\ \cdot \\ \cdot \\ \cdot \\ 0 \\ 0 \\ 0 \\ 0\end{array}\right]$                      (50)

Similarly for the point, y₂ we have,

$F_{k}^{\prime}\left(y_{2}\right)=\sum_{j=1}^{N} X_{2 j}^{*} \theta_{k}\left(y_{j}\right) \begin{gathered}\text { for } i=1,2, \ldots, N \\ k=1,2, \ldots, N\end{gathered}$                 (51)

Then again, the tri-diagonal system of equations is formed as:

$\left[\begin{array}{ccccccccc}6 & 1 & 0 & 0 & & & & & \\ 0 & 4 & 1 & 0 & \cdots & 0 & & 0 & \\ 0 & 1 & 4 & 1 & & & & & \\ & \vdots & & 0 & \ddots & 0 & & \vdots & \\ & & & & & 1 & 4 & 1 & 0 \\ & 0 & & 0 & \cdots & 0 & 1 & 4 & 0 \\ & & & & & 0 & 0 & 1 & 6\end{array}\right]\left[\begin{array}{c}X^{*}{ }_{21} \\ X^{*}{ }_{22} \\ X^{*}{ }_{23} \\ \cdot \\ \cdot \\ \cdot \\ X^{*}{ }_{2N-3} \\ X^{*}{ }_{2N-2} \\ X^{*}{ }_{2N-1} \\ X^{*}_{2N} \end{array}\right]=\left[\begin{array}{c}-3 / h \\ 3 / h \\ 0 \\ 0 \\ \cdot \\ \cdot \\ 0 \\ 0 \\ 0 \\ 0\end{array}\right]$                         (52)

We continue to find the tri-diagonal systems for the remaining y_i’s and the system for last y_N is given by:

$\left[\begin{array}{ccccccccc}6 & 1 & 0 & 0 & & & & & \\ 0 & 4 & 1 & 0 & \cdots & 0 & & 0 & \\ 0 & 1 & 4 & 1 & & & & & \\ & \vdots & & 0 & \ddots & 0 & & \vdots & \\ & & & & & 1 & 4 & 1 & 0 \\ & 0 & & 0 & \cdots & 0 & 1 & 4 & 0 \\ & & & & & 0 & 0 & 1 & 6\end{array}\right]\left[\begin{array}{c}X^{*}{ }_{N1} \\ X^{*}{ }_{N2} \\ X^{*}{ }_{N3} \\ \cdot \\ \cdot \\ \cdot \\ X^{*}{ }_{NN-3} \\ X^{*}{ }_{NN-2} \\ X^{*}{ }_{NN-1} \\ X^{*}_{NN} \end{array}\right]=\left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ \cdot \\ \cdot \\ 0 \\ 0 \\ -6 / h \\ 6 / h\end{array}\right]$                         (53)

The solution of the above systems provides the coefficients

$X_{11}^{*}, X_{12}^{*} \ldots, X_{1 N}^{*}, X_{21}^{*}, X_{22}^{*} \ldots, X_{2 N}^{*} \ldots, X_{N 1}^{*}, X_{N 2}^{*} \ldots, X_{N N}^{*}$.

Then the values of $Y^{*}{ }_{i j}$ for $i=1,2,3 \ldots N, j=1,2,3 \ldots N$ are calculated as follows:

$\left.\begin{array}{c}Y_{i j}^{*}=2 X^{*}{ }_{i j}\left(X^{*}{ }_{i j}-\frac{1}{y_{i}-y_{j}}\right) & \text { for } i \neq j \\ Y^{*}{ }_{i i}=-\sum_{i=1, i \neq j}^{N} Y^{*}{ }_{i j} & i=j\end{array}\right\}$                       (54)

The reduced system of ordinary differential equations in time, that is, represented as for i=1, 2, 3…, N.

$V_{t}=R_{m}\left(U_{1}, U_{2}, T_{1}, T_{2}, N_{1}, N_{2} U_{p}, T_{p}\right)$                    (55)

The system is solved by the following strong stability preserving scheme (section 3.1).

3.1 Computation of $u_{1}, u_{2}, \mathbf{N}_{1}, N_{2}, T_{1}, T_{2}$

To obtain the velocity, micro-rotation, and temperature profiles for MHD (magnetohydrodynamic) Couette flow of micropolar and micropolar dusty fluids in the respective Zone, replace the approximation of the spatial components of the I and II order derivatives obtained by using MCB-DQM (modified cubic b-spline differential quadrature method). Hence the system of coupled partial Eqns. (17)-(24) is numerically solved with the initial and boundary conditions (12)-(15) and the linear velocities, angular velocity component (microrotation), and temperature profiles of both fluids and particles are readily obtained. The Eqns. (17)-(24) can be updated as follows:

Zone-I (Micropolar fluid: −k ≤ y ≤ 0).

$\frac{\partial U_{1}}{\partial t}=\frac{1}{R e} \cdot\left[\begin{array}{c}\eta_{1} \sum_{j=1}^{N} X^{*}{ }_{i j} N_{1}\left(y_{j}, t\right)+\left(1+\eta_{1}\right)\left(\sum_{j=1}^{N} Y^{*}{ }_{i j} U_{1}\left(y_{j}, t\right)\right) \\ -\frac{H a^{2}(1+B i . B e)}{(1+B i . B e)^{2}+B e^{2}} U_{1}\left(y_{j}, t\right)\end{array}\right]$                         (56)

$\frac{\partial N_{1}}{\partial t}=\frac{1}{R e}\left[\left(1+\frac{\eta_{1}}{2}\right)\left(\sum_{j=1}^{N} Y^{*}{ }_{i j} N_{1}\left(y_{j}, t\right)\right)-\eta_{1}\left(2 N_{1}\left(y_{j}, t\right)+\sum_{j=1}^{N} X^{*}{ }_{i j} U_{1}\left(y_{j}, t\right)\right)\right]$                   (57)

$\frac{\partial T_{1}}{\partial t}=\frac{E c}{R e}\left[\begin{array}{c}\frac{1}{P r . E c}\left(\sum_{j=1}^{N} Y^{*}{ }_{i j} T_{1}\left(y_{j}, t\right)\right)+\left(\sum_{j=1}^{N} X^{*}{ }_{i j} U_{1}\left(y_{j}, t\right)\right)^{2}+ \\ \eta_{1}\left(2 N_{1}\left(y_{j}, t\right)+\sum_{j=1}^{N} X^{*}{ }_{i j} U_{1}\left(y_{j}, t\right)\right)^{2}+\delta_{1}\left(\sum_{j=1}^{N} X^{*}{ }_{i j} N_{1}\left(y_{j}, t\right)\right)^{2} \\ \frac{H a^{2}(1+B i . B e)}{(1+B i . B e)^{2}+B e^{2}} U_{1}\left(y_{j}, t\right)^{2}\end{array}\right]$                 (58)

Zone-II (Micropolar dusty fluid: 0 ≤ y ≤ k)

$\frac{\partial U_{2}}{\partial t}=\frac{r_{1}}{r_{2} \cdot R e}\left[\begin{array}{c}\eta_{2} \sum_{j=1}^{N} X^{*}{ }_{i j} N_{2}\left(y_{j}, t\right)+\left(1+\eta_{2}\right)\left(\sum_{j=1}^{N} Y^{*}{ }_{i j} U_{2}\left(y_{j}, t\right)\right) \\ -\frac{H a^{2}(1+B i . B e)}{r_{1}\left((1+B i . B e)^{2}+B e^{2}\right)} U_{2}\left(y_{j}, t\right)-R\left(U_{2}\left(y_{j}, t\right)-U_{p}\left(y_{j}, t\right)\right)\end{array}\right]$                          (59)

$\frac{\partial U_{p}}{\partial t}=\frac{R \cdot r_{1} \cdot r_{3}}{r_{2} \cdot R e}\left[U_{2}\left(y_{j}, t\right)-U_{p}\left(y_{j}, t\right)\right]$                  (60)

$\frac{\partial N_{2}}{\partial t}=\frac{r_{1}}{r_{2} \cdot R e}\left[\left(1+\frac{\eta_{2}}{2}\right)\left(\sum_{j=1}^{N} Y_{i j}^{*} N_{2}\left(y_{j}, t\right)\right)-\eta_{2}\left(2 N_{2}\left(y_{j}, t\right)+\sum_{j=1}^{N} X^{*}{ }_{i j} U_{2}\left(y_{j}, t\right)\right)\right]$                 (61)

$\left.\begin{array}{l}\frac{\partial T_{2}}{\partial t}=\frac{E c \cdot r_{1}}{R e \cdot r_{2} \cdot C_{r}}\left[\begin{array}{c}\frac{K r}{P r . E c \cdot r_{1}}\left(\sum_{j=1}^{N} Y_{i j}^{*} T_{2}\left(y_{j}, t\right)\right)+\left(\sum_{j=1}^{N} X^{*}{ }_{i j} U_{2}\left(y_{j}, t\right)\right)^{2}+\eta_{2}\left(\sum_{j=1}^{N} X_{i j}^{*} U_{2}\left(y_{j}, t\right)\right) \\ +\delta_{2}\left(\sum_{j=1}^{N} X^{*}{ }_{i j} N_{2}\left(y_{j}, t\right)\right)^{2}+\frac{H a^{2}(1+B i . B e)}{r_{1}\left((1+B i . B e)^{2}+B e^{2}\right)} U_{2}\left(y_{j}, t\right)^{2}+ \\\frac{2}{3} \frac{R \cdot K_{r}}{P r . E c \cdot r_{1}}\left(T_{p}\left(y_{j}, t\right)-T_{2}\left(y_{j}, t\right)\right)+\frac{R \cdot C_{r} \cdot r_{3}}{E c}\left(U_{2}\left(y_{j}, t\right)-U_{p}\left(y_{j}, t\right)\right)^{2}\end{array}\right.^{2}\end{array}\right]$                      (62)

$\frac{\partial T_{p}}{\partial t}=\frac{2}{3} \frac{R * k_{r} * C_{r} * r_{3}}{C_{P r} * r_{1} * P r} \frac{\left(T_{2}\left(y_{j}, t\right)-T_{p}\left(y_{j}, t\right)\right)}{R e}$                   (63)

Thus, equations are reduced into a system of ordinary differential equations in time, that is, for i=1, 2, 3…, N, and the system is solved by the robust four-step third-order SSP RK43 [33]. The velocities and microrotation in both Zones are obtained as follows:

At first - the step for i=1,2,3…, n

Zone-I ($-\mathrm{k} \leq \mathrm{y} \leq 0$) (micropolar fluid):

$U_{1_{1}}=U_{1_{0}}+\frac{\Delta t}{2}\left(\frac{1}{R e} \cdot\left[\begin{array}{c}\eta_{1} \sum_{j=1}^{N} X_{i j}^{*} N_{1_{0}}\left(y_{j}, t\right)+ \\ \left(1+\eta_{1}\right)\left(\sum_{j=1}^{N} Y_{i j}^{*} U_{1_{0}}\left(y_{j}, t\right)\right) \\ -\frac{H a^{2}(1+B i . B e)}{(1+B i . B e)^{2}+B e^{2}} U_{1_{0}}\left(y_{j}, t\right)\end{array}\right]\right)$                      (64)

$N_{1_{1}}=N_{10}+\frac{\Delta t}{2}\left(\frac{1}{R e}\left[\left(1+\frac{\eta_{1}}{2}\right)\left(\sum_{j=1}^{N} Y^{*}{ }_{i j} N_{1_{0}}\left(y_{j}, t\right)\right)-\eta_{1}\left(\begin{array}{c}2 N_{1_{0}}\left(y_{j}, t\right)+ \\ \sum_{j=1}^{N} X^{*} U_{i j} U_{1_{0}}\left(y_{j}, t\right)\end{array}\right)\right]\right)$                          (65)

$\left.T_{1_{1}}=T_{1_{0}}+\frac{\Delta t}{2}\left(\begin{array}{c}\frac{E c}{R e}\left[\begin{array}{c}\frac{1}{P r . E c}\left(\sum_{j=1}^{N} Y^{*}{ }_{i j} T_{10}\left(y_{j}, t\right)\right)+\left(\sum_{j=1}^{N} X^{*}{ }_{i j} U_{10}\left(y_{j}, t\right)\right)^{2}+\eta_{1}\left(2 N_{10}\left(y_{j}, t\right)+\sum_{j=1}^{N} X^{*}{ }_{i j} U_{1_{0}}\left(y_{j}, t\right)\right)^{2} \\ +\delta_{1}\left(\sum_{j=1}^{N} X^{*}{ }_{i j} N_{1}\left(y_{j}, t\right)\right)^{2}+\frac{H a^{2}(1+B i . B e)}{(1+B i . B e)^{2}+B e^{2}} U_{1_{0}}\left(y_{j}, t\right)^{2}\end{array}\right.\end{array}\right]\right)$                   (66)

Zone-II (0 ≤ y ≤ k) (micropolar dusty fluid):

$U_{2_{1}}=U_{2_{0}}+\frac{\Delta t}{2}\left(\frac{r_{1}}{r_{2} \cdot R e}\left[\begin{array}{c}\eta_{2} \sum_{j=1}^{N} X^{*}{ }_{i j} N_{2_{0}}\left(y_{j}, t\right)+\left(1+\eta_{2}\right)\left(\sum_{j=1}^{N} Y^{*}{ }_{i j} U_{2_{0}}\left(y_{j}, t\right)\right) \\ -\frac{H a^{2}(1+B i . B e)}{r_{1}\left((1+B i \cdot B e)^{2}+B e^{2}\right)} U_{2_{0}}\left(y_{j}, t\right)-R\left(U_{2_{0}}\left(y_{j}, t\right)-U_{p_{0}}\left(y_{j}, t\right)\right)\end{array}\right]\right)$                      (67)

$U_{p_{1}}=U_{p_{0}}+\frac{\Delta t}{2}\left(\frac{R \cdot r_{1} \cdot r_{3}}{r_{2} \cdot R e}\left[U_{20}\left(y_{j}, t\right)-U_{p_{0}}\left(y_{j}, t\right)\right]\right)$                (68)

$N_{2_{1}}=N_{1_{0}}+\frac{\Delta t}{2}\left(\frac{r_{1}}{r_{2} \cdot \operatorname{Re}}\left[\left(1+\frac{\eta_{2}}{2}\right)\left(\sum_{j=1}^{N} Y^{*}{ }_{i j} N_{2_{0}}\left(y_{j}, t\right)\right)-\eta_{2}\left(2 N_{2_{0}}\left(y_{j}, t\right)+\sum_{j=1}^{N} X^{*}{ }_{i j} U_{2_{0}}\left(y_{j}, t\right)\right)\right]\right)$                      (69)

$T_{2_{1}}=T_{1_{0}}+\frac{\Delta t}{2} \cdot\left(\frac{E c . r_{1}}{R e \cdot r_{2} \cdot C_{r}}\left[\begin{array}{c}\frac{K r}{\operatorname{Pr} . E c . r_{1}}\left(\sum_{j=1}^{N} Y^{*} T_{2_{0}}\left(y_{j}, t\right)\right)+\left(\sum_{j=1}^{N} X^{*}{ }_{i j} U_{2_{0}}\left(y_{j}, t\right)\right)^{2} \\ +\eta_{2}\left(2 N_{2_{0}}\left(y_{j}, t\right)+\sum_{j=1}^{N} X^{*}{ }_{i j} U_{2_{0}}\left(y_{j}, t\right)\right)^{2}+ \\ \delta_{2}\left(\sum_{j=1}^{N} X^{*}{ }_{i j} N_{2_{0}}\left(y_{j}, t\right)\right)^{2}+\frac{H a^{2}(1+B i . B e)}{r_{1}\left((1+B i . B e)^{2}+B e^{2}\right)} U_{2_{0}}\left(y_{j}, t\right)^{2} \\ +\frac{2}{3} \frac{R \cdot K_{r}}{P r . E c \cdot r_{1}}\left(T_{p_{0}}\left(y_{j}, t\right)-T_{2_{0}}\left(y_{j}, t\right)\right)+\frac{R \cdot C_{r} \cdot r_{3}}{E c}\left(U_{2_{0}}\left(y_{j}, t\right)-U_{p_{0}}\left(y_{j}, t\right)\right)^{2}\end{array}\right]\right)$                       (70)

$T_{p_{1}}=T_{p_{0}}+\frac{\Delta t}{2}\left(\frac{2}{3} \frac{R * k_{r} * C_{r} * r_{3}}{C_{P r} * r_{1} * \operatorname{Pr}} \frac{\left(T_{2_{0}}\left(y_{j}, t\right)-T_{p_{0}}\left(y_{j}, t\right)\right)}{R e}\right)$                         (71)

At the first step of the method, the conditions (12)-(15) are regarded favourably.

At the second step for i=1,2,3…, n:

Zone-I ($-k \leq y \leq 0$) (micropolar fluid):

$\begin{gathered}

U_{1_{2}}=U_{1_{1}}+ \\

\frac{\Delta t}{2}\left(\frac{1}{R e} \cdot\left[\begin{array}{c}

\eta_{1} \sum_{j=1}^{N} X^{*}{ }_{i j} N_{1_{1}}\left(y_{j}, t\right)+ \\

\left(1+\eta_{1}\right)\left(\sum_{j=1}^{N} Y^{*}{ }_{i j} U_{1_{1}}\left(y_{j}, t\right)\right) \\

-\frac{H a^{2}(1+B i . B e)}{(1+B i \cdot B e)^{2}+B e^{2}} U_{1_{1}}\left(y_{j}, t\right)

\end{array}\right]\right)

\end{gathered}$                        (72)

$N_{12}=N_{11}+\frac{\Delta t}{2}\left(\frac{1}{R e}\left[\left(1+\frac{\eta_{1}}{2}\right)\left(\sum_{j=1}^{N} Y^{*}{ }_{i j} N_{1_{1}}\left(y_{j}, t\right)\right)-\eta_{1}\left(2 N_{1_{1}}\left(y_{j}, t\right)+\sum_{j=1}^{N} X_{i j}^{*} U_{1_{1}}\left(y_{j}, t\right)\right)\right]\right)$                    (73)

$\left.T_{1_{2}}=T_{1_{1}}+\frac{\Delta t}{2}\left(\begin{array}{c}\frac{E c}{R e}\left[\begin{array}{c}\frac{1}{P r . E c}\left(\sum_{j=1}^{N} Y^{*}{ }_{i j} T_{1_{1}}\left(y_{j}, t\right)\right)+\left(\sum_{j=1}^{N} X^{*}{ }_{i j} U_{1_{1}}\left(y_{j}, t\right)\right)^{2}+ \\ \eta_{1}\left(2 N_{1_{1}}\left(y_{j}, t\right)+\sum_{j=1}^{N} X^{*}{ }_{i j} U_{1_{1}}\left(y_{j}, t\right)\right)^{2}+\delta_{1}\left(\sum_{j=1}^{N} X^{*}{ }_{i j} N_{1_{1}}\left(y_{j}, t\right)\right)^{2} \\ \frac{H a^{2}(1+B i . B e)}{(1+B i . B e)^{2}+B e^{2}} U_{1_{1}}\left(y_{j}, t\right)^{2}\end{array}\right.\end{array}\right]\right)$                       (74)

Zone-II (0 ≤ y ≤ k) (micropolar dusty fluid):

$U_{2_{2}}=U_{2_{1}}+\frac{\Delta t}{2}\left(\frac{r_{1}}{r_{2} \cdot R e}\left[\begin{array}{c}\eta_{2} \sum_{j=1}^{N} X^{*}{ }_{i j} N_{2_{1}}\left(y_{j}, t\right)+\left(1+\eta_{2}\right)\left(\sum_{j=1}^{N} Y^{*}{ }_{i j} U_{2_{1}}\left(y_{j}, t\right)\right) \\ -\frac{H a^{2}(1+B i . B e)}{r_{1}\left((1+B i . B e)^{2}+B e^{2}\right)} U_{2_{1}}\left(y_{j}, t\right)-R\left(U_{2_{1}}\left(y_{j}, t\right)-U_{p_{1}}\left(y_{j}, t\right)\right)\end{array}\right]\right)$                        (75)

$U_{p_{2}}=U_{p_{1}}+\frac{\Delta t}{2}\left(\frac{R \cdot r_{1} \cdot r_{3}}{r_{2} \cdot \operatorname{Re}}\left[U_{21}\left(y_{j}, t\right)-U_{p_{1}}\left(y_{j}, t\right)\right]\right)$            (76)

$N_{22}=N_{11}+\left(\frac{r_{1}}{r_{2} \cdot R e}\left[\left(1+\frac{\eta_{2}}{2}\right)\left(\sum_{j=1}^{N} Y^{*}{ }_{i j} N_{21}\left(y_{j}, t\right)\right)-\eta_{2}\left(\begin{array}{c}2 N_{2_{1}}\left(y_{j}, t\right)+ \\ \sum_{j=1}^{N} X^{*}{ }_{i j} U_{21}\left(y_{j}, t\right)\end{array}\right)\right]\right)$                           (77)

$\left.T_{2_{2}}=T_{1_{1}}+\frac{\Delta t}{2} \cdot\left(\frac{E c \cdot r_{1}}{\text { Re. } r_{2} \cdot C_{r}}  \left[ \begin{array}{ccc} \frac{K r}{P r . E c \cdot r_{1}}\left(\sum_{j=1}^{N} Y^{*}{ }_{i j} T_{2_{1}}\left(y_{j^{\prime}} t\right)\right)+\left(\sum_{j=1}^{N} X^{*}{ }_{i j} U_{2_{1}}\left(y_{j^{\prime}}, t\right)\right)^{2}+\eta_{2}\left(2 N_{21}\left(y_{j^{\prime}}, t\right)+\sum_{j=1}^{N} X^{*}{ }_{i j} U_{2_{1}}\left(y_{j^{\prime}}, t\right)\right)^{2} \\ +\delta_{2}\left(\sum_{j=1}^{N} X^{*}{ }_{i j} N_{21}\left(y_{j^{\prime}}, t\right)\right)^{2}+\frac{H a^{2}(1+B i . B e)}{r_{1}\left((1+B i . B e)^{2}+B e^{2}\right)} U_{21}\left(y_{j^{\prime}}, t\right)^{2}\\+\frac{2}{3} \frac{R \cdot K_{r}}{P r \cdot E c \cdot r_{1}}\left(T_{p_{1}}\left(y_{j^{\prime}}, t\right)-T_{2_{1}}\left(y_{j^{\prime}}, t\right)\right)+\frac{R \cdot C_{r} \cdot r_{3}}{E c}\left(U_{21}\left(y_{j^{\prime}} t\right)-U_{p_{1}}\left(y_{j}, t\right)\right)^{2} \end{array}\right]  \right]\right)$                          (78)

$T_{p_{2}}=T_{p_{1}}+\frac{\Delta t}{2}\left(\frac{2}{3} \frac{R * k_{r} * C_{r} * r_{3}}{C_{P r} * r_{1} * \operatorname{Pr}} \frac{\left(T_{21}\left(y_{j}, t\right)-T_{p_{1}}\left(y_{j}, t\right)\right)}{\operatorname{Re}}\right)$                    (79)

At the second step of the method, the conditions (12)-(15) are regarded favourably.

At the third step for i=1,2,3…,n

Zone-I ( $-k \leq y \leq 0$) (micropolar fluid):

$U_{1_{3}}=\frac{2 . U_{1_{0}}}{3}+\frac{U_{1_{1}}}{3}+\frac{\Delta t}{6}\left(\frac{1}{R e} \cdot\left[\begin{array}{c}\eta_{1} \sum_{j=1}^{N} X^{*}{ }_{i j} N_{1_{2}}\left(y_{j}, t\right)+ \\ \left(1+\eta_{1}\right)\left(\sum_{j=1}^{N} Y^{*}{ }_{i j} U_{1_{2}}\left(y_{j}, t\right)\right) \\ -\frac{H a^{2}(1+B i . B e)}{(1+B i \cdot B e)^{2}+B e^{2}} U_{1_{2}}\left(y_{j}, t\right)\end{array}\right]\right)$                            (80)

$N_{1_{3}}=\frac{2 . N_{1_{0}}}{3}+\frac{N_{1_{1}}}{3}+\frac{\Delta t}{6}\left(\frac{1}{R e}\left[\begin{array}{c}\left(1+\frac{\eta_{1}}{2}\right)\left(\sum_{j=1}^{N} Y^{*}{ }_{i j} N_{12}\left(y_{j}, t\right)\right) \\ -\eta_{1}\left(\begin{array}{c}2 N_{12}\left(y_{j}, t\right)+ \\ \sum_{j=1}^{N} X^{*}{ }_{i j} U_{12}\left(y_{j}, t\right)\end{array}\right)\end{array}\right]\right)$                   (81)

$T_{1_{3}}=\frac{2 . T_{1_{0}}}{3}+\frac{T_{1_{1}}}{3}+\frac{\Delta t}{6}\left(\begin{array}{c}\left.\frac{E c}{R e}\left[\begin{array}{c}\frac{1}{P r . E c}\left(\sum_{j=1}^{N} Y_{i j}^{*} T_{12}\left(y_{j}, t\right)\right)+\left(\sum_{j=1}^{N} X_{i j}^{*} U_{12}\left(y_{j}, t\right)\right)^{2}+ \\ \eta_{1}\left(2 N_{1_{2}}\left(y_{j}, t\right)+\sum_{j=1}^{N} X^{*}{ }_{i j} U_{1_{2}}\left(y_{j}, t\right)\right)^{2} \\ +\delta_{1}\left(\sum_{j=1}^{N} X^{*}{ }_{i j} N_{1_{2}}\left(y_{j}, t\right)\right)^{2}+\frac{H a^{2}(1+B i . B e)}{(1+B i . B e)^{2}+B e^{2}} U_{1_{2}}\left(y_{j}, t\right)^{2}\end{array}\right]\right)\end{array}\right.$                     (82)

Zone-II (0 ≤ y ≤ k) (micropolar dusty fluid):

$U_{2_{3}}=\frac{2 . U_{2_{0}}}{3}+\frac{U_{2_{1}}}{3}+\frac{\Delta t}{6}\left(\frac{r_{1}}{r_{2} \cdot R e}\left[\begin{array}{c}\eta_{2} \sum_{j=1}^{N} X^{*}{ }_{i j} N_{2_{2}}\left(y_{j}, t\right)+\left(1+\eta_{2}\right)\left(\sum_{j=1}^{N} Y^{*}{ }_{i j} U_{2_{2}}\left(y_{j}, t\right)\right) \\ -\frac{H a^{2}(1+B i . B e)}{r_{1}\left((1+B i, B e)^{2}+B e^{2}\right)} U_{2_{2}}\left(y_{j}, t\right)-R\left(U_{2_{2}}\left(y_{j}, t\right)-U_{p_{2}}\left(y_{j}, t\right)\right)\end{array}\right]\right)$                        (83)

$U_{p_{3}}=\frac{2 . U_{p_{0}}}{3}+\frac{U_{p_{1}}}{3}+\frac{\Delta t}{6}\left(\frac{R \cdot r_{1} \cdot r_{3}}{r_{2}, R e}\left[U_{22}\left(y_{j}, t\right)-U_{p_{2}}\left(y_{j}, t\right)\right]\right)$               (84)

$N_{2_{3}}=\frac{2 . N_{2_{0}}}{3}+\frac{N_{2_{1}}}{3}+\frac{\Delta t}{6}\left(\frac{r_{1}}{r_{2} \cdot R e}\left[\left(1+\frac{\eta_{2}}{2}\right)\left(\sum_{j=1}^{N} Y^{*}{ }_{i j} N_{2_{2}}\left(y_{j}, t\right)\right)-\eta_{2}\left(\begin{array}{c}2 N_{2_{2}}\left(y_{j}, t\right)+ \\ \sum_{j=1}^{N} X_{i j}^{*} U_{2}\left(y_{j}, t\right)\end{array}\right)\right]\right)$                          (85)

$T_{2_{3}}=\frac{2 \cdot T_{2}}{3}+\frac{T_{2_{1}}}{3}+\frac{\Delta t}{6} \cdot\left(\frac{E c \cdot r_{1}}{R e \cdot r_{2} \cdot C_{r}} \left[\begin{array}{c}\frac{K r}{P r . E c \cdot r_{1}}\left(\sum_{j=1}^{N} Y^{*}{ }_{i j} T_{2_{2}}\left(y_{j}, t\right)\right)+\left(\sum_{j=1}^{N} X^{*}{ }_{i j} U_{2_{2}}\left(y_{j}, t\right)\right)^{2} \\ +\eta_{2}\left(2 N_{2_{2}}\left(y_{j}, t\right)+\sum_{j=1}^{N} X^{*}{ }_{i j} U_{2_{2}}\left(y_{j}, t\right)\right)^{2}+ \\ \delta_{2}\left(\sum_{j=1}^{N} X^{*}{ }_{i j} N_{2_{2}}\left(y_{j}, t\right)\right)^{2}+\frac{H a^{2}(1+B i . B e)}{r_{1}\left((1+B i \cdot B e)^{2}+B e^{2}\right)} U_{2_{2}}\left(y_{j}, t\right)^{2} \\ +\frac{2}{3} \frac{R \cdot K_{r}}{P r \cdot E c \cdot r_{1}}\left(T_{p_{2}}\left(y_{j}, t\right)-T_{2_{2}}\left(y_{j}, t\right)\right)+\frac{R \cdot C_{r} r_{3}}{E c}\left(U_{2_{2}}\left(y_{j}, t\right)-U_{p_{2}}\left(y_{j}, t\right)\right)^{2}\end{array}\right]  \right)$                     (86)

$T_{p_{3}}=\frac{2 \cdot T_{p_{0}}}{3}+\frac{T_{p_{1}}}{3}+\frac{\Delta t}{6} \cdot\left(\frac{2}{3} \frac{R * k_{r} * C_{r} * r_{3}}{C_{P r} * r_{1} * P r} \frac{\left(T_{2_{1}}\left(y_{j}, t\right)-T_{p_{1}}\left(y_{j}, t\right)\right)}{R e}\right)$                         (87)

At the third step of the method, the conditions (12)-(15) are once again regarded favourably.

At the fourth step for i=1,2,3…,n:

Zone-I ($-k \leq y \leq 0$,) (micropolar fluid):

$U_{1_{4}}=U_{1_{3}}+\frac{\Delta t}{2}\left(\frac{1}{R e} \cdot\left[\begin{array}{c}\eta_{1} \sum_{j=1}^{N} X^{*}{ }_{i j} N_{1_{3}}\left(y_{j}, t\right)+ \\ \left(1+\eta_{1}\right)\left(\sum_{j=1}^{N} Y_{i j}^{*} U_{1_{3}}\left(y_{j}, t\right)\right) \\ -\frac{H a^{2}(1+B i . B e)}{(1+B i \cdot B e)^{2}+B e^{2}} U_{1_{3}}\left(y_{j}, t\right)\end{array}\right]\right)$                          (88)

$N_{1_{4}}=N_{1_{3}}+\frac{\Delta t}{2}\left(\frac{1}{R e}\left[\begin{array}{c}\left(1+\frac{\eta_{1}}{2}\right)\left(\sum_{j=1}^{N} Y^{*}{ }_{i j} N_{1_{3}}\left(y_{j}, t\right)\right) \\ -\eta_{1}\left(\begin{array}{c}2 N_{1_{3}}\left(y_{j}, t\right)+ \\ \sum_{j=1}^{N} X^{*} U_{1_{3}}\left(y_{j}, t\right)\end{array}\right)\end{array}\right]\right)$                     (89)

$T_{1_{4}}=T_{1_{3}}+\frac{\Delta t}{2}\left(\frac{E c}{R e}\left[\begin{array}{c}\frac{1}{\operatorname{Pr} . E c}\left(\sum_{j=1}^{N} Y^{*}{ }_{i j} T_{13}\left(y_{j}, t\right)\right)+\left(\sum_{j=1}^{N} X^{*}{ }_{i j} U_{13}\left(y_{j}, t\right)\right)^{2}+ \\ \eta_{1}\left(2 N_{1_{3}}\left(y_{j}, t\right)+\sum_{j=1}^{N} X^{*}{ }_{i j} U_{1_{3}}\left(y_{j}, t\right)\right)^{2} \\ +\delta_{1}\left(\sum_{j=1}^{N} X^{*}{ }_{i j} N_{1_{3}}\left(y_{j}, t\right)\right)^{2}+\frac{H a^{2}(1+B i . B e)}{(1+B i . B e)^{2}+B e^{2}} U_{1_{3}}\left(y_{j}, t\right)^{2}\end{array}\right]\right)$                    (90)

Zone-II (0 ≤ y ≤ k) (micropolar dusty fluid):

$U_{2_{4}}=U_{2_{3}}+\frac{\Delta t}{2}\left(\frac{r_{1}}{r_{2} \cdot R e}\left[\begin{array}{c}\eta_{2} \sum_{j=1}^{N} X_{i j}^{*} N_{2_{3}}\left(y_{j}, t\right)+ \\ \left(1+\eta_{2}\right)\left(\sum_{j=1}^{N} Y_{i j}^{*} U_{2_{3}}\left(y_{j}, t\right)\right) \\ -\frac{H a^{2}(1+B i . B e)}{r_{1}\left((1+B i \cdot B e)^{2}+B e^{2}\right)} U_{2_{3}}\left(y_{j}, t\right) \\ -R\left(U_{2_{3}}\left(y_{j}, t\right)-U_{p_{3}}\left(y_{j}, t\right)\right)\end{array}\right]\right)$               (91)

$U_{p_{4}}=U_{p_{3}}+\frac{\Delta t}{2}\left(\frac{R \cdot r_{1} \cdot r_{3}}{r_{2} \cdot R e}\left[U_{2_{3}}\left(y_{j}, t\right)-U_{p_{3}}\left(y_{j}, t\right)\right]\right)$                   (92)

$N_{2_{4}}=N_{2_{3}}+\frac{\Delta t}{2}\left(\frac{r_{1}}{r_{2} \cdot R e}\left[\begin{array}{c}\left(1+\frac{\eta_{2}}{2}\right)\left(\sum_{j=1}^{N} Y^{*}{ }_{i j} N_{2}\left(y_{j}, t\right)\right) \\ -\eta_{2}\left(2 N_{2_{3}}\left(y_{j}, t\right)+\sum_{j=1}^{N} X_{i j}^{*} U_{23}\left(y_{j}, t\right)\right)\end{array}\right)\right]$                (93)

$T_{2_{4}}=T_{2_{3}}+\frac{\Delta t}{2} \cdot\left(\frac{E c \cdot r_{1}}{R e \cdot r_{2} \cdot C_{r}}  \left[\begin{array}{c}\frac{K r}{P r . E c \cdot r_{1}}\left(\sum_{j=1}^{N} Y_{i j}^{*} T_{2_{3}}\left(y_{j}, t\right)\right)+\left(\sum_{j=1}^{N} X^{*}{ }_{i j} U_{23}\left(y_{j}, t\right)\right)^{2} \\ +\eta_{2}\left(2 N_{2_{3}}\left(y_{j}, t\right)+\sum_{j=1}^{N} X^{*}{ }_{i j} U_{2_{3}}\left(y_{j}, t\right)\right)^{2}+\delta_{2}\left(\sum_{j=1}^{N} X^{*}{ }_{i j} N_{2_{3}}\left(y_{j}, t\right)\right)^{2} \\ +\frac{H a^{2}(1+B i . B e)}{r_{1}\left((1+B i . B e)^{2}+B e^{2}\right)} U_{2_{3}}\left(y_{j}, t\right)^{2}+\frac{2}{3} \frac{R \cdot K_{r}}{\operatorname{Pr} \cdot \operatorname{Ec} \cdot r_{1}}\left(\begin{array}{c}T_{p_{3}}\left(y_{j}, t\right) \\ -T_{2_{3}}\left(y_{j}, t\right)\end{array}\right)+\frac{R \cdot C_{r} \cdot r_{3}}{E c}\left(\begin{array}{c}U_{2_{3}}\left(y_{j}, t\right) \\ -U_{p_{3}}\left(y_{j}, t\right)\end{array}\right)^{2}\end{array}\right] \right)$                          (94)

$T_{p_{4}}=T_{p_{3}}+\frac{\Delta t}{2}\left(\frac{2}{3} \frac{R * k_{r} * C_{r} * r_{3}}{C_{P r} * r_{1} * \operatorname{Pr}} \frac{\left(T_{2_{3}}\left(y_{j}, t\right)-T_{p_{3}}\left(y_{j}, t\right)\right)}{\operatorname{Re}}\right)$                      (95)

At the fourth step of the method, the conditions (12)-(15) are also regarded favourably.

Hence the fluid (linear) velocity and angular velocity (Eringen microrotation) and temperatures profiles i. e. $U_{1}, N_{1}, T_{1}$, of micropolar fluid in Zone-I, and also the fluid velocity (linear) and particle velocity components, angular velocity, and temperature profiles. $U_{2}, U_{p} N_{2}, T_{2}, T_{p}$ for micropolar dusty fluid in Zone-II can be numerically obtained at the fourth step of MCB-DQM.

3.2 Skin friction coefficient

As fluid moves over the plates, it creates friction on their edges, which inhibits forward motion and causes skin friction drift on the surface. This impact is measured using the skin friction coefficient. The expression of skin friction coefficients at both plates are calculated as

At lower plate,

$\left(\boldsymbol{C}_{f}\right)_{y=-1}=\frac{2}{\boldsymbol{R e}}\left[\left(1+\eta_{1}\right) \frac{\partial U_{1}}{\partial y}+\eta_{1} N_{1}\right]_{y=-1}$                    (96)

At upper plate,

$\left(C_{f}\right)_{y=1}=\frac{2 r_{1}}{r_{2} R e}\left[\left(1+\eta_{1}\right) \frac{\partial U_{1}}{\partial y}+\eta_{1} N_{1}\right]_{y=1}$                    (97)

3.3 Nusselt number

The expression of the Nusselt number at both plates is calculated as:

At lower plate,

$\left(N_{U}\right)_{y=-1}=\left(-\frac{\partial T_{1}}{\partial y}\right)_{y=-1}$                     (98)

At upper plate,

$\left(N_{U}\right)_{y=1}=\left(-\frac{\partial T_{2}}{\partial y}\right)_{y=1}$                        (99)

Two immiscible non-Newtonian magnetohydrodynamic micropolar and micropolar dusty fluids are considered in the horizontal duct with heat transfer. The flow is induced by upper plate movement (Couette flow). Modified cubic B-spline differential quadrature method (MCB-DQM) is applied to get the numerical results for velocity, Microrotation, and temperature profiles of fluid and particle phase. The effects of important fluid parameters have been identified. The main outcomes of the current study are summarized as

1. The Modified cubic B-spline differential quadrature method is in good agreement with obtained numerical results in the limiting case of fluid flow, with the exact solution.
2. Fluid velocities, microrotation, and temperature profiles of fluids and particles are accelerating with time.
3. The micropolar parameters η₁ and η₂ of both fluids affects the velocities and temperatures.
4. The Hall (Be) and Ion-slip (Bi) parameters are increasing the velocity, and lessening the temperature of both fluids. The angular velocity of micropolar dusty and micropolar fluids is found to be in decreasing nature.
5. Hartmann number (Ha²) is inversely proportional to the velocities of the fluid and particle phases.
6. Reynold’s number (Re) affects the fluid and particle velocities and their temperatures with an inversely proportional relation. While if an external pressure is applied then it changes its nature decreasing to increasing.
7. Dust particle and fluid temperatures increase with an enhancement in Eckert number (Ec), the ratio of viscosities r₁, the raito of densities r₂.